Abstract

While the problems of image coding and audio coding have frequently
been assumed to have similarities, specific sets of relationships
have remained vague. One area where there should be a meaningful
comparison is with central masking noise estimates, which
define the codec's quantizer step size.
In the past few years, progress has been made on this problem
in the auditory domain (Allen and Neely, J. Acoust. Soc. Am.,
{\bf 102}, 1997, 3628-46; Allen, 1999, Wiley Encyclopedia of
Electrical and Electronics Engineering, Vol. 17, p. 422-437,
Ed. Webster, J.G., John Wiley \& Sons, Inc, NY).
It is possible that some useful insights might now be obtained
by comparing the auditory and visual cases.
In the auditory case it has been shown, directly from psychophysical
data, that below about 5 sones
(a measure of loudness, a unit of psychological intensity),
the loudness JND is proportional to the square root of the loudness
$\DL(\L) \propto \sqrt{\L(I)}$.
This is true for both wideband noise and tones, having
a frequency of 250 Hz or greater.
Allen and Neely interpret this to mean that the internal noise is
Poisson, as would be expected from neural point process noise.
It follows directly that the Ekman fraction (the relative loudness JND),
decreases as one over the square root of the loudness, namely
$\DL/\L \propto 1/\sqrt{\L}$.
Above ${\L} = 5$ sones, the relative loudness JND
$\DL/\L \approx 0.03$ (i.e., Ekman law).
It would be very interesting to know if this same
relationship holds for the visual case between brightness $\B(I)$
and the brightness JND $\DB(I)$. This might be tested by measuring
both the brightness JND and the brightness as a function of
intensity, and transforming the intensity JND into a brightness JND, namely
\[
\DB(I) = \B(I+ \DI) - \B(I)
\approx \DI \frac{d\B}{dI}.
\]
If the Poisson nature of the loudness relation (below 5 sones)
is a general result of central neural noise, as is anticipated,
then one would expect that it would also hold in vision,
namely that $\DB(\B) \propto \sqrt{\B(I)}$.
%The history of this problem is fascinating, starting with Weber and Fechner.
It is well documented that the exponent in the S.S. Stevens' power
law is the same for loudness and brightness (Stevens, 1961)
\nocite{Stevens61a}
(i.e., both brightness $\B(I)$ and loudness $\L(I)$ are proportional to
$I^{0.3}$). Furthermore, the brightness JND data are more like
Riesz's near miss data than recent 2AFC studies of JND measures
\cite{Hecht34,Gescheider97}.